Fully developed laminar incompressible flow in an eccentric annulus

Abstract

Past investiga Drs of 1 minar incompressible fluid flow in eccentric annular ducts have concentrated on the fully developed problem. Each investigation yielded a different expression for the fully developed velocity, which was then used to determine other information pertinent to fully developed flow. Snyder and Goldstein (1965), for example, used their expression to determine local wall shear stress and various friction factors. A summary of heat transfer and fluid flow research in eccentric annular and various other ducts has been compiled by Shah and London (1971). The purpose of this paper is to present a different representation of the fully developed velocity which proves quite useful in the analysis of the entrance region problem. A brief description of the general entrance region problem and an explanation of the role of our representation in solving this problem for an eccentric annular duct folIow.In 1964, Sparrow et al. proposed a linearized version of the entrance region problem for a straight duct with an arbitrary cross section. A method of solution, which represented the entrance region velocity as the sum of the fully developed velocity and a difference velocity, was also proposed and, in fact, applied to the parallel plate and circular tube problems. The application of their method to other geometries is warranted by the close agreement of their analytical results with experimental data.The author has recently solved the equation which governs the difference velocity in an eccentric annular duct using the Galerkin method. The coordinate functions used to represent the difference velocity are used in this paper to represent the fully developed velocity. Thus, the entrance region velocity can be expressed in terms of a single set of coordinate functions. Furthermore, our representation allows for the calculation of all Galerkin inner products, difference velocity coefficients, and fully developed velocity coefficients in closed form. 0001-1541-78-9995-0733-$00.75. 0 The American Institute of Chemical Engineers, 1978. ANALYSIS The equation governing the flow is given byThe geometry under consideration is shown in Figure 1. (1) a2u a% 1 --.+-=-ax2 ay2 cL dz where the pressure gradient d p / d z and the viscosity are constants. We impose the nonslip boundary condition u = 0 on C, the duct walls. Symmetry considerations allow us to solve Equation (1 only in, say, the lower half of the annulus. It also follows from symmetry that M a y = 0 on B, the plane of symmetry of the duct. If we define rZ2 -r12 -e2 2e r22 -r12 + e2 2e c1 = and cz= then the bipolar transformation given by 1 -2hy x2 + y2 -h2 q = arctan maps the lower half of the annulus onto the rectangle shown in Figure 2. In terms of the 4' -coordinates, Equation (1) is given by ( 2 ) a2v a2v -1 -+-= a$ &q2 [cosh(S + k2) -cos 71' where -U v=-h2 d p P d z -is the dimensionless velocity. The boundary conditions for